COMP/MATH 553 Algorithmic
Game Theory
Lecture 5: Myerson’s Optimal
Sep 17, 2014
Yang Cai
An overview of today’s class
Expected Revenue = Expected Virtual Welfare
2 Uniform [0,1] Bidders Example
Optimal Auction
Revelation Principle Recap
S1(b1 )
. . . . . .
S1( )
S2( )
. . . . . .
S2(b2 )
S n( )
Original Mechanism M
x(s1(b1), s2(b2),...,sn(bn))
New Mechanism M’
Revenue Maximization
Bayesian Analysis Model
 A single-dimensional environment, e.g. single-item
 The private valuation vi of participant i is assumed to be drawn from a distribution
Fi with density function fi with support contained in [0,vmax].
We assume that the distributions F1, . . . , Fn are independent (not necessarily
In practice, these distributions are typically derived from data, such as bids in
past auctions.
 The distributions F1 , . . . , Fn are known in advance to the mechanism designer.
The realizations v1, . . . , vn of bidders’ valuations are private, as usual.
Revenue-Optimal Auctions
 [Myerson ’81
 Single-dimensional settings
 Simple Revenue-Optimal auction
What do we mean by optimal?
 Step 0: What types of mechanism do we need to consider? In other words, optimal
amongst what mechanisms?
 Consider the set of mechanisms that have a dominant strategy equilibrium.
 Want to find the one whose revenue at the dominant strategy equilibrium is the
 A large set of mechanisms. How can we handle it?
 Revelation Principle comes to rescue! We only need to consider the directrevelation DSIC mechanisms!
Expected Revenue = Expected
Virtual Welfare
Revenue = Virtual Welfare
[Myerson ’81
] For any single-dimensional
Let F= F1 × F2 × ... × Fn be the joint value distribution, and
(x,p) be a DSIC mechanism. The expected revenue of this
Ev~F[Σi pi(v)]=Ev~F[Σi xi(v) φi (vi)],
where φi (vi) := vi- (1-Fi(vi))/fi(vi) is called bidder i’s virtual
value (fi is the density function for Fi).
Myerson’s OPTIMAL
Two Bidders + One Item
 Two bidders’ values are drawn i.i.d. from U[0,1].
 Vickrey with reserve at ½
• If the highest bidder is lower than ½, no one wins.
• If the highest bidder is at least ½, he wins the item and pay
max{1/2, the other bidder’s bid}.
 Revenue 5/12.
 This is optimal. WHY???
Two Bidders + One Item
 Virtual value for v: φ(v)= v- (1-F(v))/f(v) = v- (1-v)/1= 2v-1
 Optimize expected revenue = Optimize expected virtual welfare!!!
 Should optimize virtual welfare on every bid profile.
 For any bid profile (v1,v2), what allocation rule optimizes virtual welfare?
(φ(v1), φ(v2))=(2v1-1, 2v2-1).
If max{v1,v2} ≥ 1/2, give the item to the highest bidder
Otherwise, φ(v1), φ(v2) < 0. Should not give it to either of the two.
 This allocation rule is monotone.
Revenue-optimal Single-item Auction
 Find the monotone allocation rule that optimizes expected virtual welfare.
 Forget about monotonicity for a while. What allocation rule optimizes
expected virtual welfare?
 Should optimize virtual welfare on every bid profile v.
- max Σi xi(v) φi (vi). , s.t Σi xi(v) ≤ 1
 Call this Virtual Welfare-Maximizing Rule.
Revenue-optimal Single-item Auction
 Is the Virtual Welfare-Maximizing Rule monotone?
 Depends on the distribution.
 Definition 1 (Regular Distributions): A single-dimensional distribution F is
regular if the corresponding virtual value function v- (1-F(v))/f(v) is nondecreasing.
 Definition 2 (Monotone Hazard Rate (MHR)): A single-dimensional
distribution F has Monotone Hazard Rate, if (1-F(v))/f(v) is non-increasing.
Revenue-optimal Single-item Auction
 What distributions are in these classes?
MHR: uniform, exponential and Gaussian distributions and many more.
Regular: MHR and Power-law...
Irregular: Multi-modal or distributions with very heavy tails.
 When all the Fi’s are regular, the Virtual Welfare-Maximizing Rule is
Two Extensions Myerson did (we won’t teach)
 What if the distributions are irregular?
Point-wise optimizing virtual welfare is not monotone.
Need to find the allocation rule that maximizes expected virtual welfare among all monotone
ones. Looks hard...
This can be done by “ironing” the virtual value functions to make them monotone, and at the
same time preserving the virtual welfare.
 We restrict ourselves to DSIC mechanisms
Myerson’s auction is optimal even amongst a much larger set of “Bayesian incentive
compatible (BIC)” (essentially the largest set) mechanisms.
For example, this means first-price auction (at equilibrium) can’t generate more revenue than
Myerson’s auction.
 Won’t cover them in class.
- Section 3.3.5 in “Mechanism Design and Approximation”, book draft by Jason Hartline.
- “Optimal auction design”, the original paper by Roger Myerson.

similar documents